Plane Deformations of Incompressible Fiber-Reinforced Materials

Abstract

A continuum theory of finite, plane deformations of composites consisting of materials reinforced by strong fibers is discussed. The composite is assumed to be incompressible, and the fibers are treated as inextensible and continuously distributed. The analysis is not restricted to any particular material behavior such as elasticity, plasticity, or visco-elasticity. Plane deformations are kinematically determinate, in that the deformation can be found by using the constraint conditions and suitable displacement boundary conditions. The reactions to the constraints produce stress equilibrium in any kinematically admissible deformation. The theory admits stress singularities of an unusual kind: a single fiber or normal line can carry a finite load. Simple examples illustrating this and other points of the theory are given.

A mixed boundary-value problem for fiber-reinforced materials

“…Both of these deformations, and all others that are kinematically admissible, can be supported in equilibrium with suitable tractions on the boundary [1], However, as we shall see, the prescribed conditions of zero traction are satisfied by the deformation in Fig. 2 but not, in general, by that in Fig.…”

“…Their orthogonal trajectories, the fibers, are then parallel curves and the constraints imply [1] that the distance between any two fibers must be the same as it was initially.…”

“…In a previous paper [1] we have outlined a theory of plane deformations of materials which are both incompressible and reinforced with inextensible fibers. We showed that, under weak restrictions on the shear stress response of the material, every kinematically admissible deformation is also statically admissible.…”

“…The constraints of incompressibility and inextensibility imply [1] that normal lines must be straight when the fibers are initially parallel. Their orthogonal trajectories, the fibers, are then parallel curves and the constraints imply [1] that the distance between any two fibers must be the same as it was initially.…”

“…Assuming that the stress components <j31 and a32 are zero by symmetry, the stress for a plane deformation is of the form [1] <j = -p(I -aa) + Taa + S33kk + <S(an + na).…”